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Index

Absolute parallelism on F M, 102, 138, 141, 154, 203 on F2 M, 198,202

Adapted coframe on F2 M, 199 frame on F2 M, 199

Affine bundle, 178 subbundle, 172, 178

Algebraic model, 144, 163 Almost

contact metric manifold, 128 Hermitian manifold, 129, 212 Kahler manifold, 133, 213 symplectic manifold, 124, 153 symplectic submanifold, 152 complex structure 101 131 , , ,

149, 156, , 158 159, 160, 161, 208, 212

integrability of, 131 on Rn x gl( n, R), 157, 159

contact metric structure, 128 fundamental form of, 128

contact structure 113 128 157 , , , , 159, 160, 161, 162

normal, 131, 132, 133, 157, 162 on gl(n, R), 157, 158 on Rn, 157, 158

Hermitian structure 102 113 , , , 163, 164, 212

basis adapted to, 130 on FM, 130

presymplectic structure, 34 product structure on F2 M, 199

227

symplectic structure 34 149 , , , 150, 152

integrability of, 151 integrable or symplectic, 150,

153 Associated

fibre bundle, 200 trivialization on

FFM, 203 FF2M, 202

b-boundary, 168, 169 b-completion, 167, 168

of the 2-dimensional Friedman spacetime, 167

b-incomplete curve, 167, 169 or connection incomplete mani­

fold, 167 spacetime, 167

Bianchi identity, 184 Bracket product

map, 44 of g-valued forms, 58

Bundle of almost symplectic frames, 124 frames adapted to an almost

contact structure, 160 orthonormal frames, 122, 154 pseudo-orthonormal frames, 168

Canonical basis, 41

of Rn, 138, 145, 195 of gl( n, R), 138, 195

228

Canonical decomposition of 0(2), 195 fiat connection on F M, 103, 154

covariant derivative of the, 154 curvature tensor of the, 103 torsion tensor of the, 103

frame, 31, 32 isomorphisms, 139 identifications, 33

Chevalley-Eilenberg differential, 164 Completion of a manifold, 168 Complete

lift to F M of, a covariant differentiation, 73 a derivation, 72 a differentiable function, 50,

85 a linear connection, 63, 66, 70,

73,93 local components of, 64, 66

tensor fields, 54, 87 of type (p, V), 66

(0, 2)-tensor field, 35 (0, s )-tensor field, s ~ 0, 35, 52 (1, 1 )-tensor field, 32 (1, s )-tensor field, s ~ 2, 32, 52 vector field, 7, 9, 24, 51, 85,

110, 124 a vector-valued differentiable

function, 49 lift to F2M,

of a (1,1 )-tensor field, 215 Riemannian submanifold, 167

Conformal system of variable a-connec­

tions, 188 stability of connection incomple­

teness, 186 variations of Lorentz structure,

185 Connected component of FM, 165,

183, 185

Index

Connection, adapted to a G-structure, 68,

142, 143, 152, 153 fiat, 61, 69, 102, 105, 147 form, 60, 64, 80, 84, 173, 181

of a linear connection, 137 in a principal fibre bundle, 60 linear, 75, 76, 80, 83, 85, 93, 94,

102, 109, 110, 137, 138, 139, 140, 141, 142, 144, 145, 146, 147,148,149, 151, 153, 158, 179,180,181,201,202

structure equations of, 142 torsion of, 159, 164

map, 110 metric, 101, 154, 165, 166

on a frame bundle, 185 on a fibred manifold, 172, 173,

175, 181 opposite, 73, 83, 86 reducible, 68 tangent, 60 torsion free, 63, 99, 109, 143, 147

Cosymplectic manifold, 132, 133 fiat, 133

Curvature, 159, 164 form, 61, 141, 181, 184

of the prolongation, 61 tensor of a linear connection, 66,

83, 89, 93, 112 of the Levi-Civita connection of

the Sasaki-Mok metric, 115, 118

Ricci, 119, 121 scalar, 119, 122

constant, 122 sectional, 119, 120

bounded, 120, 134

De Rham cohomology, 184 Derivation, 71, 104

defined by a tensor field, 71

Index

Derivative covariant, 66, 86, 100, 102, 124,

205 of a section, 172

Diagonal lift to FM of

a G-structure, 77, 82 a metric, 112 a I-form, 80 tensor fields, 82 a (1, I)-tensor field, 81 a (l,s)-tensor field, s::::: 1,82 a (0,2)-tensor field, 100, 101,

107, 109 a (O,s)-tensor field, s ::::: 1,81

lift to F2 M of a I-form, 206 a (1, I)-tensor field, 207, 208,

217 a (0,2)-tensor field, 209, 217

product of Lie groups, 75 prolongation of a G-structure to

F2 M, 216, 217 Differentiable

function tensor, 139, 140 of type (p, V), 142

V -valued function, 44 of type (Pl,J~V), 49 oftype (p, V), 49

map of type (p, V), 45 Differential system, 36

completely integrable, 36 Differentiation,

covariant, 61, 63, 73, 83 intrinsic, 70

Distribution, 90 complementary, 126 G-invariant, 180 horizontal, 80, 84, 146, 154, 156,

172, 198 integrable, 90, 126, 127, 147 vertical on FM, 80, 84,146,154,

229

156, 172 vertical on F2 M, 198

Endomorphism, 141 Einstein manifold, 121, 134 Expected information metric, 187,

189

F2 M, 193,204 canonical form of, 194

Families of a-connections, 188 Fibre

coordinates, 50 bundle, 171, 174

Fibred manifold, 171, 173

connection, 181 morphism, 178 product, 178

First jet bundle

of a fibred manifold, 172, 174, 181

of a principal G-bundle, 176, 180

jet functor, 175 Form

almost symplectic, 108, 109, 151, 153, 210, 211, 212

fundamental, 163 of type (p, V), 47 of type (PI, J;V), 47

I-Form g-valued, 59 V-valued, 45 J; V-valued, 46

r-Form g-valued, 57

oftype (ad, g), 59 of type (p, g), 59

tensorial of type (ad,gl(n, R)). 140

230

J;s-valued, 58 of type (ad, J;S), 59

V-valued, 46 of type (p, V), 47 tensorial, 61

Frame bundle, 8, 18, 20, 80, 173, 177 canonical I-form of, 63, 80, 84 principal fibre bundle structure

of, 8 system of connections, 182 trivialization of the, 138

J-structure, 125, 126, 127 fundamental form of, 127 integrable, 127 partially integrable, 127, 146

JAK-manifold, 127, 128 J H -manifold, 127 JK-manifold, 127, 128 J(3, I)-structure, 146, 147, 156

framed,147 integrable, 146 normal, 147

J(3, -I)-structure, 128, 147 integrable, 147 partial integrability of, 147

J( 4, 2)-structure, 148 integrability of, 148 partial integrability, 148

J(4, -2)-structure, 148

G2(n), 193, 194 adjoint representation of, 194 Lie algebra of, 194

Gauge field theories, 183 r -transformation, 77

infinitesimal, 77 Geodesic, 70, 125, 156

complete, 168 horizontal, 125 incomplete, 167 of a linear connection, 97, 98,

99

Index

of second order, 200, 201 of the complete lift of a linear

connection, 70 of the horizontal lift of a linear

connection, 97 of the Levi-Civita connection of

the Sasaki-Mok metric, 125 spray, 125

Geometrical quantization, 185 G-invariant k-linear map, 184 G-structure, 21, 34, 76, 138, 143,

144, 149, 214, 215, 216 automorphism of, 24, 77

infinitesimal, 24, 77 canonical prolongation of, 53 defined by

(1, I)-tensor field, 138, 142, 143

(0,2)-tensor fields, 138, 139 diagonal prolongation of, 75, 77,

78 field of

coframes adapted to, 29 frames adapted to, 29, 69

first structure tensor of, 141 integrable (or fiat), 25, 27, 29, 31,

34, 141,215 canonical frame of, 25

isomorphism of, 21, 24 over F2 M, 202 O-deformable, 141 w-associated

on FM, 138, 139, 141, 149, 202

on F2M,202 polynomial, 144 prolongation of, 21, 27, 29, 31 S/(n,R)-, 37, 54, 55

Harmonic local diffeomorphisms, 113 map, 134, 135

Riemannian submersion, 134 Hermitian manifold, 132, 133, 213 Holonomy bundles with respect

Index

to the universal connection, 183

Homogeneous space, 12 Hopf-Rinow theorem, 165 Horizontal

component, 114 lift of,

a curve, 125 a covariant differentiation, 105 a derivation, 104 a distribution, 90 a linear connection, 92, 94, 98,

101 local components of, 95

vector field, 62, 76, 85, 110, 111, 129, 142

tensor fields, 87 (1, I)-tensor field, 87, 89, 92,

100 I-form, 88

subspace, 151, 199, 200

Incomplete manifold, 165 Induced basis, 41 Inextensible

curve, 165, 185 inco~plete curv o 165, 186

Infinitesimal automorphism, 109,211 affine transformation, 109

Inner product, 153, 163 standard on Rn, 165

Integral curve, of a sta.ndard horizontal vector

field, 125 Integral manifold, 127, 147, 152 Isomorphic vector bundles, 41 Isotropy group, 11, 30, 33, 34, 52, 53,

13~ 139, 149,214,217

Jacobi vector field, 70, 71 Jet

I-jet, 3, 53, 54 of local diffeomorphism, 8

2-jet of local diffeomorphism, 193

J~G, 9,13 Lie group structure of, 9 unit element of, 9

J;G,193 Lie group structure of, 193

J~g, 13 Lie algebra structure of, 13 structure constants of, 13

J~M, 3, 5 functorial properties of, 5 manifold structure of, 4

J;M, 192 functorial properties of, 192 manifold structure of, 192

J1Rn 12 p ,

PRn 213 p ,

vector space structure of, 213 J~V, 12

vector space structure of, 12

K1q,-curvature identity, 132 Kahler

form, 130, 163, 164, 212 manifold, 102, 133

flat, 102

231

Killing vector field, 109, 110, 124,211 with vanishing se,cond covariant

derivative, 211

Levi-Civita b-completion, 168 Levi-Civita connection, 101, 112,

127, 128, 154, 155, 211, 213 of the Sasaki metric, 113 of the Sasaki-Mok metric, 112,

113 curva.ture of, 115, 118

232

geodesic of, 125 of GD , 108

Lie algebra homomorphism, 72, 105 of vector fields on F M, 112, 129 of the affine real group, 156 of the orthogonal group, 155 structure of Rn x gl(n, R), 156

Lie derivative, 71, 91, 108, 109 Linear holonomy group, 69 Local diffeomorphism, 174 Lorentz

group, 168 manifolds, 187

Massless Klein-Gordon scalar field, 185

Metric flat, 101 Hermitian, 101, 102, 156, 212 locally Euclidean, 101, 126, 128 of Lorentz type, 168

Model spaces, 149 Model vector space, 139

Natural prolongation of a G-structu­re to F2 M, 214

Nearly Kahler manifold, 133 Nijenhuis

tensor, 126, 157, 208 of a tensor field, 92

torsion, 157, 160, 161, 164 of a tensor field of type (1,1),

131, 141

Orthogonal complement, 123 frames, 122 group, 122

Orthonormal basis, 121, 122

Parallel (1, I)-tensor field, 142 Parallelizable manifold, 102, 144

Index

Parametric model, 187 statistical model, 188

Polynomial structure, 32, 89, 144, 145, 149, 208

Presymplectic structure, 34 Principal

connection, 180, 182 fibre bundle, 17, 18, 193

reduced, 68 G-bundle, 172, 174, 180, 182

category of, 175 Probability distribution function,

187 Product bundle, 178 Projection

operators, 126 tensor, 90

Prolongation of, a connection, 60 differentiable function, 49 g-valued I-form, 60 tensorial form, 62 vector-valued forms, 46, 47

Pseudo-Riemannian manifold, 165, 168

connection incomplete, 186 metric, 186

Relativistic singularities, 185 Representation

adjoint, 59 canonical, 51, 52, 215

linear, 42, 53, 138, 139, 149, 213

induced, 43, 58, 59 linear, 39, 45, 49, 52, 58, 61

trivial, 50 Riemannian

manifold, 112, 113, 122, 125, 126, 134

flat, 120, 121, 134, 167

local diffeomorphism of, 134 locally Euclidean, 119, 127 locally symmetric, 119

Index

metric, 101, 108, 109, 110, 112, 153, 154, 163, 209, 211

adapted to an j-structure, 127 an j(3, 1 )-structure, 156

structure, 149, 153

S2(n),194 Sasaki-Mok metric, 108, 112 Sasaki metric, 108 Second order connection, 196,

200, 201, 202, 203, 204, 211, 216, 217

admissible, 201, 202 curvature form of, 196 local components of, 198 partially fiat, 212 structure equation of, 197 torsion form of, 196

Section canonical, 64 induced, 64 natural, 64 of the surjection, 173

Set of connections on a principal bundle, 176

Simply connected manifold, 144 Skewness tensor of expectation, 188 Spacetime, 165

completion, 168 manifold,167 singularities, 168, 169 singularity theory, 185

Sprays, 183 Stability of incompleteness, 169

in parametric models, 189 Standard contact structure on Rn,

158

Structural polynomial, 144, 145, 147, 147, 148, 149, 208

Structure of connections, 178 Surjection, 177 Surmersion, 171 Symplectic

subspace, 151 structure, 34, 153

System

233

stability of connections-incom­pleteness, 186

of a-connections, 188 of connections, 188

on a principal G-bundle, 178 on a fibred manifold, 178 linear, 179, 182 principal, 179

of variable a-connections, 189 space of a system of connections,

178

Tangent bundle, 7, 173 of p2-velocities, 192

of Rn, 212 of second order, 200

Target map, 54 Tensor fields

associated to differentiable ten­sors, 140

O-deformable, 140 Torsion

form of a linear connection, 63, 141

tensor of a linear connection, 66, 83, 93, 143

Totally geodesic map, 134, 135 2-frame, 193, 196

Uniform stability of completion-in­completeness, 186

Universal connection, 171, 182, 184

234

for the system of linear connec­tions, 184

linear, 185 on the tangent bundle, 183

fibre manifold connection, 181 frame bundle connection, 182 holonomy group, 183 principal bundle connection, 180

Vanishing second covariant deriva­tive, 98, 110

Vector bundle, 41 associated, 178

Vector subbundle, horizontal, 172 vertical, 172, 178

Vector field onFM

fundamental, 82, 84, 85, 98, 99, 109, 137

standard horizontal, 82, 99, 125, 137

on F2M fundamental, 194, 200, 211

corresponding to functions, 203, 204, 205

standard horizontal, 196, 200, 202 integral curve of, 201

Vertical component, 114 lift to F M of a vector field, 7,

111,129 vector I-form, 173 subspace

on FM, 151 on F2M, 194

Volume form, 54

Weil's theorem on characteristic classes, 184

Whitney sums, 41

Index